Problem: $\dfrac{ -3t + 10u }{ 2 } = \dfrac{ -4t - 10v }{ -10 }$ Solve for $t$.
Multiply both sides by the left denominator. $\dfrac{ -3t + 10u }{ {2} } = \dfrac{ -4t - 10v }{ -10 }$ ${2} \cdot \dfrac{ -3t + 10u }{ {2} } = {2} \cdot \dfrac{ -4t - 10v }{ -10 }$ $-3t + 10u = {2} \cdot \dfrac { -4t - 10v }{ -10 }$ Multiply both sides by the right denominator. $-3t + 10u = 2 \cdot \dfrac{ -4t - 10v }{ -{10} }$ $-{10} \cdot \left( -3t + 10u \right) = -{10} \cdot 2 \cdot \dfrac{ -4t - 10v }{ -{10} }$ $-{10} \cdot \left( -3t + 10u \right) = 2 \cdot \left( -4t - 10v \right)$ Distribute both sides $-{10} \cdot \left( -3t + 10u \right) = {2} \cdot \left( -4t - 10v \right)$ ${30}t - {100}u = -{8}t - {20}v$ Combine $t$ terms on the left. ${30t} - 100u = -{8t} - 20v$ ${38t} - 100u = -20v$ Move the $u$ term to the right. $38t - {100u} = -20v$ $38t = -20v + {100u}$ Isolate $t$ by dividing both sides by its coefficient. ${38}t = -20v + 100u$ $t = \dfrac{ -20v + 100u }{ {38} }$ All of these terms are divisible by $2$ $t = \dfrac{ -{10}v + {50}u }{ {19} }$